Spiral roller machine

ABSTRACT

Rolling the linear strip of single and laminated (multilayer) plates to produce a spiral shell, constituting part or whole wall of a cylinder and tubular element, is a procedure performed by using the forms and moulds. 
     The stringers (beams) of round or curved stairs and their handrails are examples of spiral shells and elements constituting part of cylinder wall. 
     Pipes with spiral joints are the example of spiral shell constituting whole wall of a cylinder and tubular element. 
     This patent provides the design of a spiral roller machine and its relevant calculations to adjust the machine, and the procedure to roll spiral shells and elements according to the required shape.

This invention relates to a machine which rolls the linear strip of single and laminated (multilayer) plates to produce a spiral shell, constituting part or whole wall of a cylinder.

The stringers (beams) of round or curved stairs and their handrails made of laminated plywoods or woods are the spiral shells and elements which form part of cylinder wall (DWG-16 a).

Common wood stair has two beams on two sides, which carry the load of the stair to the end supports. These beams are called “stringers”. For round and curved stairs, the stringers are usually made of laminated plywoods. These stringers have a complicated shape with three dimensional bent and twist. In order to roll the stringer, a form consisting of many vertical elements (called jig) which constitute the horizontal image of stair, are used. The layers of plywoods after applying carpentry glue are leaned on this form with proper slope and tightened by clamps. After glue cured and hardened, the stringers are separated from the form and the consequent shape remains. This method is a hard process and takes lots of man-hour and expense.

On the other hand, pipes with spiral joints are the tubular elements in which the spiral shell forms the whole wall of a cylinder (DWG-16 b). In order to roll the plate for pipe, many moulds have to be used to shape the plate.

The new idea is rolling the spiral shells or elements by a roller machine on longitudinal axe which makes the process easier, faster, with less expense. To reach this aim there were three major challenges:

-   -   1) Theoretical converting of real coordinate and geometry to         rotated coordinate on longitudinal axe of work-piece, which is         solved by geometric analysis.     -   2) The rolling axe is not perpendicular to movement direction of         work-piece. This problem is solved by using special multi         rollers system.     -   3) Predicting the change in curve of work-piece after exit from         end roller, and amending the applied curvature, which is solved         by mechanics of material theory.

Theory:

Every spiral shell consists of four longitudinal edge curves with the same function. Finding the function of one edge is adequate to define the whole function of the spiral shell.

If we consider the roller as FIG. 1 of DWG-A:

“A”: Location of start roller

“B”: Location of end roller

“R₁”: The radius of spiral image on plane perpendicular to the cylinder axe

“α₁”: Exterior angle between tangent of start and end roller image on plane perpendicular to the cylinder axe

“α_(S)”: The angle of spiral edge with the plane perpendicular to the cylinder axe

“V”: The vector from start roller to end roller

$\overset{\_}{V}\begin{matrix} {{\Delta \; X} = {{R_{1}{Cos}\; \alpha_{1}} - R_{1}}} \\ {{\Delta \; Y} = {R_{1}{Sin}\; \alpha_{1}}} \\ {{\Delta \; Z} = {R_{1}\alpha_{1}{Tan}\; \alpha_{S}}} \end{matrix}$ V ²=(ΔX ² +ΔY ² +ΔZ ²)

V ²=(R ₁ ² Cos² α₁ +R ₁ ²−2R ₁ ² Cos α₁ +R ₁ ² Sin² α₁ +R ₁ ²α₁ ² Tan² α_(S))

V ² =R ₁ ²(Cos² α₁+1−2 Cos α₁+Sin² α₁+α₁ ² Tan² α_(S))

, Cos² α₁+Sin² α₁=1

V ² =R ₁ ²(2−2 Cos α₁+α₁ ² Tan² α_(S))

$\begin{matrix} {\left( \frac{V}{R_{1}} \right)^{2} = {2 - {2{Cos}\; \alpha_{1}} + {\alpha_{1}^{2}{Tan}^{2}\alpha_{S}}}} & \; \\ {{Must}\mspace{14mu} {be}\mspace{14mu} {solved}\mspace{14mu} {by}\mspace{14mu} {iterative}\mspace{14mu} {method}} & \; \\ {\alpha_{1} = \frac{\sqrt{\left( \frac{V}{R_{1}} \right)^{2} - 2 + {2{Cos}\; \alpha_{1}}}}{{Tan}\; \alpha_{S}}} & {{Formula}\mspace{14mu} 1} \end{matrix}$

Between the start and end roller, each spiral edge is rolled approximately as a circle.

“R₂”: Radius of spiral edge in roller machine

“α₂”: Exterior angle between the tangent of start and end roller on spiral edge

Considering FIG. 1 of DWG-A for spiral edge we have:

${V = {2\; R_{2}{Sin}\frac{\alpha_{2}}{2}}},{\alpha_{2} = {\left. \frac{S_{2}}{R_{2}}\Rightarrow V \right. = {\left. {2\; R_{2}{Sin}\frac{S_{2}}{2\; R_{2}}}\Rightarrow R_{2} \right. = \frac{V}{2\; {Sin}\frac{S_{2}}{2\; R_{2}}}}}}$ ${S_{2} = \frac{S_{1}}{{Cos}\; \alpha_{S}}},{S_{1} = {\left. {\alpha_{1} \times R_{1}}\Rightarrow R_{2} \right. = \frac{V}{2\; {Sin}\frac{\alpha_{1} \times R_{1}}{2\; R_{2}{Cos}\; \alpha_{S}}}}}$

If we consider:

required R=R_(R)=R₂, we have:

“R_(R)”: Required output radius of spiral edge

$\begin{matrix} \begin{matrix} \begin{matrix} {{Required}\mspace{14mu} {radius}\mspace{14mu} {of}\mspace{14mu} {spiral}\mspace{14mu} {edge}} \\ {{in}\mspace{14mu} {the}\mspace{14mu} {plane}\mspace{14mu} {of}\mspace{14mu} {machine}} \end{matrix} \\ {{Must}\mspace{14mu} {be}\mspace{14mu} {solved}\mspace{14mu} {by}\mspace{14mu} {iterative}\mspace{14mu} {method}} \end{matrix} & \; \\ {R_{R} = \frac{V}{2\; {Sin}\frac{\alpha_{1} \times R_{1}}{2\; R_{R}{Cos}\; \alpha_{S}}}} & {{Formula}\mspace{14mu} 2} \\ {{``\Delta_{R}"}\text{:}\mspace{14mu} {Required}\mspace{14mu} {output}\mspace{14mu} \Delta \mspace{14mu} {of}\mspace{14mu} {spiral}\mspace{14mu} {edge}} & \; \\ {{Required}\mspace{14mu} {shift}\mspace{14mu} {of}\mspace{14mu} {middle}\mspace{14mu} {roller}\mspace{14mu} {in}\mspace{14mu} {machine}} & \; \\ {\Delta_{R} = {R_{R} - \sqrt{R_{R}^{2} - \left( \frac{V}{2} \right)^{2}}}} & {{Formula}\mspace{14mu} 3} \end{matrix}$

Roll Back:

In all single and laminated plates, between the rollers, bending force is applied. When the work-piece exits from end roller, and consequently the force is eliminated, a part of this curvature will roll back and the final curvature will be less than what it was in machine. To amend this change, we should predict the amount of this roll back and roll the spiral with more curvature so that the output curvature becomes the same as required.

When we roll the single layer plate, the rolling process is in plastic limit of material and according to the theory of plasticity we have:

“ξ”: Strain of plate at far edge

“ξ_(y)”: Strain of plate corresponding to yield stress

“t”: Thickness of plate

“R_(A)”: Adjusted radius of spiral edge in machine

$\begin{matrix} {{\frac{S + {\Delta \; S}}{S} = \frac{R_{A}}{R_{A} - {t/2}}},{\frac{S + {\Delta \; S} - {\Delta S}^{\prime}}{S} = \frac{R_{R}}{R_{R} - {t/2}}}} \\ {\frac{\Delta \; S^{\prime}}{S} = {\left. \xi_{Return}\Rightarrow{\frac{R_{A}}{R_{A} - {t/2}} - \xi_{Return}} \right. = \frac{R_{R}}{R_{R} - {t/2}}}} \end{matrix}$

As an approximation in structural steel plate in existing plastic field we can consider:

$\begin{matrix} \begin{matrix} {\xi_{Return} = \left. {1.3\; \xi_{y}}\Rightarrow{\frac{R_{A} - R_{A} + {t/2}}{R_{A} - {t/2}} - {1.3\; \xi_{y}}} \right.} \\ {= {{\frac{R_{R} - R_{R} + {t/2}}{R_{R} - {t/2}}\frac{t}{{2\; R_{A}} - t}} - {1.3\; \xi_{y}}}} \\ {= \left. \frac{t}{{2\; R_{R}} - t}\Rightarrow\frac{t}{{2\; R_{A}} - t} \right.} \\ {= {\frac{t}{{2\; R_{R}} - t} + {\frac{1.3\; \xi_{y}}{t}2\; R_{A}} - t}} \\ {= \left. \frac{1}{\frac{1}{{2\; R_{R}} - t} + \frac{1.3\; \xi_{y}}{t}}\Rightarrow{2\; R_{A}} \right.} \\ {= {\frac{1}{\frac{1}{{2\; R_{R}} - t} + \frac{1.3\; \xi_{y}}{t}} + t}} \end{matrix} & \; \\ \begin{matrix} {{Adjusted}\mspace{14mu} {radius}\mspace{14mu} {on}\mspace{14mu} {machine}\mspace{14mu} {in}\mspace{14mu} {order}\mspace{14mu} {to}\mspace{14mu} {amend}} \\ {{the}\mspace{14mu} {roll}\mspace{14mu} {back}\mspace{14mu} {for}\mspace{14mu} {single}\mspace{14mu} {layer}\mspace{14mu} {plate}} \end{matrix} & \; \\ {R_{A} = {\frac{1}{\frac{1}{R_{R} - {t/2}} + \frac{2.6\; \xi_{y}}{t}} + {t/2}}} & {{Formula}\mspace{14mu} 4} \\ {{As}\mspace{14mu} {figure}\mspace{14mu} 3\mspace{14mu} {of}\mspace{14mu} D\; W\; G\text{-}A\text{:}} & \; \\ \begin{matrix} {{Adjustment}\mspace{14mu} {of}\mspace{14mu} {rollers}\mspace{14mu} {in}\mspace{14mu} {order}\mspace{14mu} {to}\mspace{14mu} {amend}} \\ {{the}\mspace{14mu} {roll}\mspace{14mu} {back}\mspace{14mu} {for}\mspace{14mu} {single}\mspace{14mu} {layer}\mspace{14mu} {plate}} \end{matrix} & \; \\ {\Delta_{A} = {R_{A} - \sqrt{R_{A}^{2} - \left( \frac{V}{2} \right)^{2}}}} & {{Formula}\mspace{14mu} 5} \end{matrix}$

On the other hand for laminated plates the rolling process is usually in elastic limit of material.

When we roll the laminated plates, the layers slide freely on each other. After rolling, we have to restrain layers from sliding back, by using screws, weld, combination of glue and temporary screws or clamps parallel to the rollers.

According to the mechanics of material theory we have:

“I_(A)”: Moment of Inertia with free sliding of layers “Δ_(A)”: Deflection with free sliding of layers (Adjustment of rollers in machine)

“I_(Return)”: Moment of Inertia with restrained sliding of layers

“Δ_(Return)”: Deflection with restrained sliding (returned “Δ” after exit from end roller)

$\begin{matrix} {{I_{A} = \frac{{nbt}^{3}}{12}}\mspace{14mu} {I_{Return} = \frac{{b({nt})}^{3}}{12}}{\frac{I_{Return}}{I_{A}} = {\frac{12\; {bn}^{3}t^{3}}{12\; {bnt}^{3}} = n^{2}}}\mspace{14mu} {I_{Return} = {n^{2}I_{A}}}{\Delta_{A} = \frac{{PV}^{3}}{48\; {EI}_{A}}}\mspace{14mu} {\Delta_{Return} = \frac{{PV}^{3}}{48\; {EI}_{Return}}}\begin{matrix} {\Delta_{Remained} = {\Delta_{A} - \Delta_{Return}}} \\ {= {\frac{{PV}^{3}}{48\; {EI}_{A}} - \frac{{PV}^{3}}{48\; {EI}_{Return}}}} \\ {= {\frac{{PV}^{3}}{48\; {EI}_{A}} - \frac{{PV}^{3}}{48\; {EI}_{A}n^{2}}}} \end{matrix}{\Delta_{Remained} = {\frac{\left( {n^{2} - 1} \right){PV}^{3}}{n^{2} \times 48\; {EI}_{A}} = {\frac{n^{2} - 1}{n^{2}}\Delta_{A}}}}{{\Delta_{Remained} = \Delta_{R}},{\Delta_{R} = {\left. {\frac{n^{2} - 1}{n^{2}}\Delta_{A}}\Rightarrow\Delta_{A} \right. = {\frac{n^{2}}{n^{2} - 1}\Delta_{R}}}},{\Delta_{R} = \left( {R_{R} - \sqrt{R_{R}^{2} - \left( \frac{V}{2} \right)^{2}}} \right)}}} & \; \\ \begin{matrix} {{Adjustment}\mspace{14mu} {of}\mspace{14mu} {rollers}\mspace{14mu} {in}\mspace{14mu} {order}\mspace{14mu} {to}\mspace{14mu} {amend}} \\ {{the}\mspace{14mu} {roll}\mspace{14mu} {back}\mspace{14mu} {for}\mspace{14mu} {laminated}\mspace{14mu} {plates}} \end{matrix} & \; \\ {\Delta_{A} = {\frac{n^{2}}{n^{2} - 1}\left( {R_{R} - \sqrt{R_{R}^{2} - \left( \frac{V}{2} \right)^{2}}} \right)}} & {{Formula}\mspace{14mu} 6} \\ {{As}\mspace{14mu} {figure}\mspace{14mu} 3\mspace{14mu} {of}\mspace{14mu} D\; W\; G\text{-}A\text{:}} & \; \\ \begin{matrix} {R_{A}^{2} = \left. {\left( {R_{A} - \Delta_{A}} \right)^{2} + \left( \frac{V}{2} \right)^{2}}\Rightarrow R_{A}^{2} \right.} \\ {= {R_{A}^{2} + \Delta_{A}^{2} - {2\; R_{A}\Delta_{A}} + {V^{2}/4}}} \end{matrix} & \; \\ \begin{matrix} {{Adjusted}\mspace{14mu} {radius}\mspace{14mu} {on}\mspace{14mu} {machine}\mspace{14mu} {in}\mspace{14mu} {order}\mspace{14mu} {to}\mspace{14mu} {amend}} \\ {{the}\mspace{14mu} {roll}\mspace{14mu} {back}\mspace{14mu} {for}\mspace{14mu} {laminated}\mspace{14mu} {plates}} \end{matrix} & \; \\ {R_{A} = \frac{\Delta_{A}^{2} + {V^{2}/4}}{2\; \Delta_{A}}} & {{Formula}\mspace{14mu} 7} \end{matrix}$

Angle Adjustment of Rollers:

From bottom to the top, the rolled plate on line “C” (in FIG. 5 of DWG-A) will be flat. To find the angle of line “C” which is in fact the rollers alignment, we have:

$\begin{matrix} {\left. \left. \begin{matrix} {{{Figure}\mspace{14mu} 5\mspace{14mu} {of}\mspace{14mu} D\; W\; G\text{-}A\text{:}\mspace{14mu} S} = {C\; {Sin}\; \alpha_{S}}} \\ {{{Figure}\mspace{14mu} 6\mspace{14mu} {of}\mspace{14mu} D\; W\; G\text{-}A\text{:}\mspace{14mu} S} = {R_{A} \times \theta}} \end{matrix} \right\}\Rightarrow\theta_{Rad} \right. = \frac{C\; {Sin}\; \alpha_{S}}{R_{A}}} & {{Formula}\mspace{14mu} 8} \\ {{{Figure}\mspace{14mu} 6\mspace{14mu} {of}\mspace{14mu} D\; W\; G\text{-}A\text{:}\mspace{14mu} X} = {R_{A}{Tan}\; \theta}} & {{Formula}\mspace{14mu} 9} \\ {{\left. \left. \begin{matrix} {{{{{Formula}\mspace{14mu} 8}\&}\mspace{11mu} 9\; \text{:}\mspace{14mu} X} = {R_{A}{{Tan}\left( \frac{C\; {Sin}\; \alpha_{S}}{R_{A}} \right)}}} \\ {{{Figure}\mspace{14mu} 7\mspace{14mu} {of}\mspace{14mu} D\; W\; G\text{-}A\text{:}\mspace{14mu} {Sin}\; \phi} = {{\frac{X}{C}\mspace{14mu} X} = {C\; {Sin}\; \phi}}} \end{matrix} \right\}\Rightarrow{C\; {Sin}\; \phi} \right. = {R_{A}{{Tan}\left( \frac{C\; {Sin}\; \alpha_{S}}{R_{A}} \right)}}},{{{Sin}\; \phi} = {\frac{R_{A}}{C}{{Tan}\left( \frac{C\; {Sin}\; \alpha_{S}}{R_{A}} \right)}}}} & \; \\ \begin{matrix} {{``\phi"}\mspace{14mu} {is}\mspace{14mu} {the}\mspace{14mu} {angle}\mspace{14mu} {of}\mspace{14mu} {rollers}\mspace{14mu} {with}\mspace{14mu} {the}\mspace{14mu} {perpendicular}} \\ {{line}\mspace{14mu} {to}\mspace{14mu} {the}\mspace{14mu} {plates}\mspace{14mu} {of}\mspace{14mu} {roller}\mspace{14mu} {machine}\mspace{14mu} {\left( {{See}\mspace{14mu} D\; W\; G\text{-}4} \right).}} \end{matrix} & \; \\ {\phi = {{arc}\; {{Sin}\left( {\frac{R_{A}}{C}{{Tan}\left( \frac{C\; {Sin}\; \alpha_{S}}{R_{A}} \right)}} \right)}}} & {{Formula}\mspace{14mu} 10} \end{matrix}$

The Machine:

There are 23 drawing pages attached to this document as following:

-   -   DWG-1: 3-D view of Roller Machine     -   DWG-2: Side section of Roller Machine     -   DWG-3: Front section of Roller Machine     -   DWG-4: Front section with adjusted angle     -   DWG-5: Gear box details     -   DWG-6: Aparted Rollerset     -   DWG-7: Details of Roller     -   DWG-8: Details of pin connection     -   DWG-9: Top view of top plate (26)     -   DWG-10: Detailed top view of top plate (26)     -   DWG-11: Top view of bottom plate (25)     -   DWG-12: Detailed top view of bottom plate (25)     -   DWG-13: Section of bottom and top plates     -   DWG-14: Roller Machine on operation     -   DWG-15: The clamp (to be used in gluing method, instead of         temporary screws)     -   DWG-16: Rolled Spiral and Cylinder     -   DWG-17: Aparted Rollerset (variant 1)     -   DWG-18: Details of Roller (variant 1)     -   DWG-19: Aparted Rollerset (variant 2)     -   DWG-20: Details of Roller (variant 2)     -   DWG-21: Mechanical adjustment instead of hydraulic jack     -   DWG-22: Machine with vertical sliding top plate     -   DWG-A: FIGS. 1 to 7 to be used in conjunction with “DESCRIPTION”

According to drawings, this machine comprises of following elements:

-   -   (1) & (3): Side (start and end) rollersets     -   (2): Middle rollerset     -   (4): Front support of top plate (rotates by gear box)     -   (5) & (6): Rear supports of top plate     -   (7): Gear box for rotating the supports and consequently         displacing the top plate     -   (8): Gear box handle     -   (9): Feeding roller (to drive the work-piece)     -   (10): Adjusting and indicating scroll for side rollersets on         bottom plate     -   (11): Adjusting and indicating scroll for middle rollerset on         bottom plate     -   (12): Tongued “U” shape drawer, supporting the side rollersets         on bottom plate     -   (13): Tongued “O” shape drawer, supporting the middle rollerset         on bottom plate     -   (14): Adjusting and indicating scroll for side rollersets on top         plate     -   (15): Adjusting and indicating scroll for middle rollerset on         top plate     -   (16): Tongued “U” shape drawer, supporting the side rollersets         on top plate     -   (17): Tongued “O” shape drawer, supporting the middle rollerset         on top plate     -   (18): Hydraulic jack for adjusting the output alignment of         rolled spiral     -   (19): Pedal controlling the movement speed and direction of         work-piece     -   (20): Pedal controlling the hydraulic jack for output alignment         of rolled spiral     -   (21): Hydraulic rotor for feeding roller     -   (22): Hydraulic pump and reservoir     -   (23): Grooved part for sliding drawers     -   (24): Shaft of feeding roller     -   (25): Bottom plate     -   (26): Top plate     -   (27): Bottom pin connection of rollersets     -   (28): Top pin connection of rollersets

The machine has two plates (25) & (26) which are connected by three support elements (4), (5) & (6). Each support is connected to plates by one tightening bolt at each end, and the distance between the top and bottom bolts of all supports are the same (“C” as DWG-6). By rotating the supports, two plates will have proportional displacement and in all conditions they will remain parallel (DWG-4). The front support is connected to bottom plate via a gear box (7), so that the supports can be rotated by the handle (8) of gear box, which consequently causes the displacement of top plate. The angle can be measured by an indicator connected to the plate or gearbox and the support. After adjusting the angle, the bolts of the supports will be tightened so that further rotation and movement be restrained.

Each plate has two drawers (12), (13), (16) & (17) which can move back and forth by sliding on tongued and grooved longitudinal parts. The drawers are tongued, and the grooved part (23) is welded to the plate (DWG-13). This movement and its calibration will be performed by one scroll connected to each drawer (10), (11), (14) & (15). The “O” shape drawers (13) & (17) each has one hole in the middle and the “U” shape drawers (12) & (16) each has two holes (one on each side). These holes are provided for pin connections (27) & (28) of rollersets (DWG-6 & DWG-8).

Three rollersets (1), (2) & (3), which each set consists of many rollers connected to two roller mounting parts and as a louver when the angle of supports (4), (5) & (6) changes and causing the change of angle between rollersets and the plates of machine (25) & (26), the rollers alignment remain parallel to the plates. Both ends of rollersets have pin connections (27) sited in sliding drawers holes on top and bottom plates. So the situation of the rollersets will be adjusted by adjusting the drawers and plates. The distance between the top and bottom end pins of each roller mounting part of rollersets must be the same as the supports of top plate (“C” as DWG-6).

A feeding roller (9) on bottom and/or top plate activated by hydraulic system drives the work-piece back and forth. Direction and speed of movement will be controlled by a pedal (19).

In order to adjust the alignment of output rolled work-piece for more convenience of process, a jack (18) adjusts the transversal angle of plates with horizon and this hydraulic jack will be controlled by an other pedal (20).

Both pedals (19) & (20) will control the hydraulic valves which each has one input, two outlets for two opposite directions of fluid flow, and one return outlet connection.

A hydraulic pump and its reservoir (22) will provide the hydraulic pressure.

Practical Summary:

In order to roll a work-piece as a spiral constituting a part or whole wall of a cylinder with radius of “R₁” and constant “Run” and “Rise” by spiral roller machine we have:

“V”: Centre to centre distance between side rollersets (See DWG-11)

“C”: Distance between the top and bottom end pins of rollerset. (See DWG-6)

$\begin{matrix} {\alpha_{S} = {{arc}\; {{Tan}\left( \frac{Rise}{Run} \right)}\mspace{14mu} {Or}\mspace{14mu} {slope}\mspace{14mu} {of}\mspace{14mu} {spiral}}} & \; \\ {{Must}\mspace{14mu} {be}\mspace{14mu} {solved}\mspace{14mu} {by}\mspace{14mu} {iterative}\mspace{14mu} {method}} & \; \\ {\alpha_{1} = \frac{\sqrt{\left( \frac{V}{R_{1}} \right)^{2} - 2 + {2\; {Cos}\; \alpha_{1}}}}{{Tan}\; \alpha_{S}}} & {{Formula}\mspace{14mu} 1} \\ {{Must}\mspace{14mu} {be}\mspace{14mu} {solved}\mspace{14mu} {by}\mspace{14mu} {iterative}\mspace{14mu} {method}} & \; \\ {R_{R} = \frac{V}{2\; {Sin}\frac{\alpha_{1}R_{1}}{2\; R_{R}{Cos}\; \alpha_{S}}}} & {{Formula}\mspace{14mu} 2} \end{matrix}$

For single layer work-piece:

$\begin{matrix} {R_{A} = {\frac{1}{\frac{1}{R_{R} - {t/2}} + \frac{2.6\; \xi_{y}}{t}} + {t/2}}} & {{Formula}\mspace{14mu} 4} \\ {{Adjustment}\mspace{14mu} {for}\mspace{14mu} {single}\mspace{14mu} {layer}\mspace{14mu} {work}\text{-}{piece}} & \; \\ {\Delta_{A} = {R_{A} - \sqrt{R_{A}^{2} - \left( \frac{V}{2} \right)^{2}}}} & {{Formula}\mspace{14mu} 5} \end{matrix}$

For laminated work-piece:

$\begin{matrix} {{Adjustment}\mspace{14mu} {for}\mspace{14mu} {laminated}\mspace{14mu} {work}\text{-}{piece}} & \; \\ {\Delta_{A} = {\frac{n^{2}}{n^{2} - 1}\left( {R_{R} - \sqrt{R_{R}^{2} - \left( \frac{V}{2} \right)^{2}}} \right)}} & {{Formula}\mspace{14mu} 6} \\ {R_{A} = \frac{\Delta_{A}^{2} + {V^{2}/4}}{2\; \Delta_{A}}} & {{Formula}\mspace{14mu} 7} \end{matrix}$

For all single and laminated work-pieces:

“Δ_(A)”: Shift of side rollers as DWG-11 adjusted by scroll (10) and (14)

$\begin{matrix} {\phi = {{arc}\; {{Sin}\left( {\frac{R_{A}}{C}{{Tan}\left( \frac{C\; {Sin}\; \alpha_{S}}{R_{A}} \right)}} \right)}}} & {{Formula}\mspace{14mu} 10} \end{matrix}$

“φ”: Angle of rollers as DWG-4 adjusted by handle (8)

Step 1: Adjust scrolls (10) & (14) according to Formula 5 (for single layer work-piece) or Formula 6 (for laminated work-piece).

Step 2: Adjust handle (8) according to Formula 10.

Step 3: For laminated work-piece apply glue to layers of work-piece.

Step 4: Put one end among all three rollersets.

Step 5: Adjust the bottom of middle rollerset (2) for most effective driving of work-piece by scroll (11). (This action tightens the work-piece between middle rollerset (2) and feeding roller (9).)

Step 6: Adjust the scroll (15) at the same condition as scroll (11).

Step 7: For laminated work-piece screw and attach the layers together from back of the work-piece at proper distance (depending on curvature) between the middle and end rollersets (2) & (3). The series of screws must be parallel to the rollersets.

Step 8: Start driving the work-piece by pushing pedal (19).

Step 9: For laminated work-piece at proper distance stop and screw the layers. Repeat this process to the end of work-piece.

Instead of screw, we can use the clamp shown on DWG-15, beside the end rollerset.

Step 10: For more convenience the condition of output rolled work-piece can be adjusted by pedal (20).

Step 11: For work-piece constituting the whole wall of a cylinder (i.e. pipes) a welding or joining machine can be installed on the outlet of end rollerset to weld or join the spiral shell continuously.

Step 12: For laminated work-piece while putting the work-piece on floor, keep the curvature on radius “R_(R)” of Formula 2. Wait the glue to be cured and hardened. Then take out the temporary screws or clamps.

The Spiral is Rolled as Required Shape Variants and Options:

1) Rollerset (Variant 1)

As an alternative of rollerset shown in DWG-6 & DWG-7, we can use rollerset as DWG-17 & DWG-18. This rollerset consists of some roller balls instead of ordinary rollers. In this rollerset, since the balls rotate on all axes, rotating the roller base is not required and the louver system is eliminated. Only one roller mounting part is adequate to install the rollers on it.

2) Rollerset (Variant 2)

As another alternative of rollerset shown in DWG-6 & DWG-7, the same roller is used but with one pin in centre of roller base connecting to one roller mounting part, and the louver system is eliminated (DWG-19 & DWG-20). When the work-piece starts moving, the roller bases rotate and become arranged in movement alignment.

3) Mechanical Adjustment for Transversal Angle of Bottom Plate

As an alternative of hydraulic jack system (18) & (20), we can adjust the angle by increasing or decreasing the length of treated bolt by an adjusting wheel installed instead of hydraulic Jack (See DWG-21).

4) Electrical Feeder

As an alternative of hydraulic feeding system (9), (19), (21) and (24), we can use electrical feeder installed on bottom and/or top plate. Also the feeder can be installed between the bottom and top plate on the centre line of work-piece.

5) Roll Bearing on Surface of Bottom and Top Plate

To facilitate the movement of work-piece, we can install roll bearings flush with the face of bottom and top plate.

6) Vertical Movement of Top Plate

By providing sliding grove for top tightening bolt on support elements of top plate and eliminating the continuous rollersets, and using three rollers on each bottom and top plate installed on drawers (12), (13), (16), and (17), we can slide the top plate in support alignment and adjust it according to the width of work-piece (See DWG-22). In this variant the work-piece is engaged among three rollers of bottom plate and three rollers of top plate at bottom and top of work-piece respectively.

7) Automated Operating System

As an option for designed machine, a hardware system programmed by software based on presented theory can be used to apply and control all the adjustments and operating procedure. 

1. A machine which rolls the linear strip of single and laminated plates to produce stringer, handrail, pipe with spiral joints, and any spiral shell and profile on which the axe of bent is not perpendicular to the longitudinal line of work-piece, constituting part or whole wall of a cylinder, comprising rollersets which each rollerset consist of at least two rollers on a common straight axe passing through the centre of rollers pivot and these axes of rollersets always remain parallel and make an angle φ with the perpendicular line to the plates of roller machine, as defined in “Description” and/or “Drawings”.
 2. The rollerset which comprises two parallel roller mounting parts, top and bottom pin connection, rollers having two pins connecting one pin to each roller mounting part, and their required connections, which acts like a louver, as defined in “Description” and/or “Drawings”.
 3. In machine of claim 1, the rollerset of claim
 2. 4. In machine of claim 1, variant 1 of rollersets comprising one roller mounting part, pin connection, and roller balls connecting to the roller mounting part, and their required connections, as defined in “Description” and/or “Drawings”.
 5. In machine of claim 1, variant 2 of rollersets comprising one roller mounting part, pin connection, and rollers connecting by one pin to the roller mounting part, and their required connections, as defined in “Description” and/or “Drawings”.
 6. In machine of claim 1, the rollerset comprising the rollers connecting to the drawers of top and bottom plates, and the top plate can move in the alignment of the support of top plate by moving its tightening bolts in the provided grove of support element, in order to adjust the vertical distance between top and bottom parallel plates according to the width of work-piece designate to roll, as defined in “Description” and/or “Drawings”.
 7. In machine of claim 1, the jack or adjusting wheel system which adjusts the transversal angle of top and bottom plates with horizon in order to ease the output alignment of work-piece, as defined in “Description” and/or “Drawings”.
 8. In machine of claim 1, the gearbox and angle indicator connecting to top or bottom plate and the support of top plate which shifts the top plate to adjust and indicate the angle of rollersets with top and bottom plates, as defined in “Description” and/or “Drawings”.
 9. In machine of claim 1, the sliding drawers system and their scroll and shift indicator connecting to the top and bottom plates which shift the rollersets and indicates the amount of the shift, as defined in “Description” and/or “Drawings”.
 10. In machine of claim 1, the automated operating system in which programmed hardware adjusts, operates, and controls all the mentioned rolling procedure, as defined in “Description”.
 11. In machine of claim 1, the hydraulic system comprising liquid reservoir, pump, rotor, pedals, valves, jack and connections which provides the motion of moving devices, as defined in “Description” and/or “Drawings”.
 12. In machine of claim 1, the feeding roller connecting to the rotor, which drives the work-piece back and forth, as defined in “Description” and/or “Drawings”.
 13. In machine of claim 1, the electric feeder which drives work-piece and can be used as another variant of hydraulic feeding system, as defined in “Description”.
 14. In machine of claim 1, the supports of top plate and their tightening screws which release or restrict the rotation of rollersets and consequently the proportional shift of plates, as defined in “Description” and/or “Drawings”.
 15. In machine of claim 1, the body which supports the plates and main part of machine.
 16. In machine of claim 1, the welding and joining machine to weld or join the spiral joints in spiral shell constituting the whole wall of cylinder as defined in “Description”.
 17. In machine of claim 1, the clamp applied to tighten the layers of laminated work-piece as defined in “Description” and/or “Drawings”.
 18. in machine of claim 1, any combination of members, variants, and options of claims 2 to
 17. 